• At maturity, protective put portfolio (stock + put) is same as cash + call
• Cash = strike
• Stock ends up \(\Rightarrow\)
  • With stock + put, keep stock
  • With cash + call, exercise, using cash, to get stock
• Stock ends down \(\Rightarrow\)
  • With stock + put, exercise, using stock, to get cash
  • With call + cash, keep cash

Prices of puts and calls

• Prior to maturity, for European options on a stock that does not pay dividends prior to the option expirations,

Example

• One year to maturity, interest rate = 5%, strike = 100
• Suppose underlying is at 90 and price of call = 10
• Price of put should be

Difference between call and put prices

• call price - put price = stock price - PV of strike
  • difference in prices (call minus put) goes up 1-for-1 with stock price
  • call price > put price when stock price > PV of strike
• put price - call price = PV of strike - price of underlying
  • difference in prices (put minus call) goes up 1/(1+r) for 1 with strike
  • e.g., strike \(\uparrow\) 5 \(\Rightarrow\) put - call \(\uparrow\) 5/(1+r)

Apple example on Feb 10, 2023

last AAPL price was $149.68

Apple put prices minus call prices

Dividends and early exercise

• Put-call parity is based on values at expiration being equal
• It may not hold for American options, because the optimal exercise conditions may be different for “stock + put” compared to “cash + call”
• It will not hold if the stock pays dividends prior to option expiration, because “stock + put” will get the dividends and “cash + call” will not.